Finite Group Databasebeta

← #13043560

Subgroups — #13043560

Order 1280.

642 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10617530C211yesmaximal, characteristic, complemented
2#10617404C211yesmaximal, characteristic, complemented
3#10610720C211yesmaximal, characteristic
4C2×#64932C211yesmaximal, characteristic, complemented
5C2×#64898C211yesmaximal, characteristic, complemented
6#10604370C211yesmaximal, characteristic, complemented
7#10604366C211yesmaximal, characteristic, complemented
8#10604000C214yesmaximal, complemented
9#10603999C214yesmaximal, complemented
10#66156C411yescharacteristic
11C2×#4094C2211yescharacteristic
12C2×#4053C2211yescharacteristic
13#65791—22no—
14#65791C2214yescomplemented
15C2×#3998—22no—
16C2×#3998C2211yescharacteristic
17C2×#3998C2211yescharacteristic
18C2×#3998C2211yescharacteristic
19C22×#674C2211yescharacteristic
20#65725—22no—
21#65725C2214yescomplemented
22D5⋊M32—22no—
23C5×#421C411yescharacteristic
24#65413C411yescharacteristic
25#65294C411yescharacteristic
26#65286C2211yescharacteristic
27#65113C2214yes—
28#65112C414yes—
29#64932C2212yescomplemented
30#64932C2212yescomplemented
31#64898C2212yescomplemented
32#64898C2212yescomplemented
33#64861C2214yescomplemented
34#64859C2212yescomplemented
35C4×#674C2212yescomplemented
36#33442—51nomaximal, Sylow
37C2×C5×#41C2×C411yescharacteristic
38C2×#709C2×C411yescharacteristic
39C2×C5×M16C2×C411yescharacteristic
40C5×#117C2×C411yescharacteristic
41#4097C812yes—
42#4096C2×C414yes—
43#4095C812yes—
44#4094—21no—
45#4094—21no—
46#4094C2×C412yes—
47#4094C2312yes—
48#4088C812yes—
49#4078C812yes—
50C2×#681—21no—
51C2×#681C2×C411yescharacteristic
52C2×#681C2×C411yescharacteristic
53C22×#168—21no—
54C22×#168C2311yescharacteristic
55#4053—21no—
56#4053—21no—
57#4053C2×C412yes—
58#4053C2312yes—
59C2×#676—21no—
60C2×C8×D5—21no—
61#3998—21no—
62#3998—21no—
63#3998—24no—
64#3998D412yes—
65#3998D412yes—
66#3998C2312yes—
67#3998C2312yes—
68C2×#674—21no—
69C2×#674—21no—
70C2×#674—21no—
71C2×#674—21no—
72C2×#674—24no—
73C2×#674C2311yescharacteristic
74C2×#674C2311yescharacteristic
75C2×#674C2312yes—
76#3990—24no—
77D5⋊C16—24no—
78C5×#104C2×C414yes—
79#3954C2×C414yes—
80#3937C2×C414yes—
81#3935C2×C412yes—
82C4×#168C2312yes—
83#3407—51no—
84#3407—51no—
85#2955—51no—
86#2171—54no—
87#2170—54no—
88#2153—51no—
89C2×#350—51no—
90C2×#350—51no—
91#2108—51no—
92C5×#41—21no—
93C5×#41—21no—
94C5×#41C2×C812yes—
95C5×#41C22×C412yes—
96C2×C5×Q8C2×C811yescharacteristic
97C2×C5×D4—21no—
98C2×C5×D4C2×C811yescharacteristic
99C22×C4×C5—21no—
100C22×C4×C5C22×C411yescharacteristic
101C2×C22⋊D5—21no—
102#709—21no—
103#709—21no—
104#709C2×C812yes—
105#709C22×C412yes—
106C2×D20C2×C811yescharacteristic
107C2×C4×D5—21no—
108C2×Dic10C2×C811yescharacteristic
109C5×M16—21no—
110C5×M16C22×C412yes—
111C2×C5×C8—21no—
112C5×C4⋊C4—21no—
113C5×C22⋊C4—21no—
114C42×C5C22×C412yes—
115#689C2×C814yes—
116#688C2×C814yes—
117#687C2×C814yes—
118#686C2×C814yes—
119#681—21no—
120#681—22no—
121#681—22no—
122#681C2×C812yes—
123#681C2×C812yes—
124#681C22×C412yes—
125C2×#168—21no—
126C2×#168—22no—
127C2×#168—22no—
128C2×#168C22×C411yescharacteristic
129C2×#168C22×C412yes—
130C2×#168C2×D411yescharacteristic
131C2×#168C2×D411yescharacteristic
132C2×#168#4111yescharacteristic
133C2×#168#4111yescharacteristic
134C2×#168C2411yescharacteristic
135#676—22no—
136#676—22no—
137C8×D5—22no—
138C8×D5—22no—
139#674—22no—
140#674—22no—
141#674—21no—
142#674—21no—
143#674C2×D414yes—
144#575—51no—
145C2×#131—51no—
146C2×#131—51no—
147#504—54no—
148#504—54no—
149#504—102no—
150#504—102no—
151C2×M32—51no—
152C2×M32—51no—
153C2×M32—51no—
154C2×M32—102no—
155C2×M32—102no—
156C22×C16—51no—
157#444—54no—
158#443—54no—
159#430—51no—
160#421—51no—
161#421AGL(1,𝔽5)11yescharacteristic
162#405—51no—
163#364—54no—
164#350—52no—
165#350—52no—
166#350—52no—
167#350—52no—
168#346—52no—
169C4×C16—52no—
170C23×C5—21no—
171C23×C5#13111yescharacteristic
172C22×D5—21no—
173C5×Q8—21no—
174C5×Q8C22×C812yes—
175C5×D4—22no—
176C5×D4—22no—
177C5×D4—21no—
178C5×D4C22×C812yes—
179C2×C4×C5—21no—
180C2×C4×C5—22no—
181C2×C4×C5—22no—
182C2×C4×C5—22no—
183C2×C4×C5C4×D411yescharacteristic
184C2×C4×C5C4×D411yescharacteristic
185C2×C4×C5C22×C811yescharacteristic
186C2×C4×C5C22×C811yescharacteristic
187C2×C4×C5#13111yescharacteristic
188C2×C4×C5C23×C411yescharacteristic
189C22⋊D5—22no—
190C22⋊D5—22no—
191C2×Dic5—21no—
192D20—21no—
193D20C22×C812yes—
194C4×D5—22no—
195C4×D5—22no—
196Dic10—21no—
197Dic10C22×C812yes—
198C5×C8—22no—
199#168—22no—
200#168C4×D411yescharacteristic
201#168C4×D411yescharacteristic
202#168C22×C814yes—
203#168C22×D411yescharacteristic
204#168C2×#4111yescharacteristic
205C2×#41—51no—
206C2×#41C2×AGL(1,𝔽5)11yescharacteristic
207#137—52no—
208#137—52no—
209AGL(1,ℤ/8)—52no—
210AGL(1,ℤ/8)—52no—
211#135—54no—
212#131—52no—
213#131—52no—
214#131—52no—
215#131—52no—
216#131—101no—
217#131—101no—
218#131—101no—
219#131—101no—
220C2×M16—51no—
221C2×M16—51no—
222C2×M16—101no—
223C2×M16—101no—
224C2×M16C2×AGL(1,𝔽5)11yescharacteristic
225C22×C8—51no—
226C22×C8—101no—
227C22×C8—101no—
228#117C2×AGL(1,𝔽5)11yescharacteristic
229M32—52no—
230M32—52no—
231M32—52no—
232M32—52no—
233M32—104no—
234M32—104no—
235M32—101no—
236M32—101no—
237C2×C16—52no—
238C2×C16—51no—
239C2×C16—51no—
240C2×C16—104no—
241C2×C16—104no—
242C2×C16—101no—
243C2×C16—101no—
244C2×C16—101no—
245C2×C16—101no—
246#108—54no—
247#104—54no—
248#104C2×AGL(1,𝔽5)14yes—
249#97—52no—
250C4×C8—52no—
251C22×C5—22no—
252C22×C5—41no—
253C22×C5—81no—
254C22×C5C8×D411yescharacteristic
255C22×C5C8×D411yescharacteristic
256C22×C5C2×#13111yescharacteristic
257C2×D5—22no—
258C2×D5—81no—
259C4×C5—22no—
260C4×C5—42no—
261C4×C5C8×D411yescharacteristic
262C4×C5C8×D411yescharacteristic
263C4×C5C2×C4×D411yescharacteristic
264C4×C5C23×C811yescharacteristic
265Dic5—22no—
266#41—21no—
267#41—21no—
268#41—52no—
269#41—52no—
270#41—101no—
271#41—101no—
272#41D5⋊C812yes—
273#41C22×AGL(1,𝔽5)12yes—
274C2×Q8—51no—
275C2×Q8D5⋊C811yescharacteristic
276C2×D4—21no—
277C2×D4—51no—
278C2×D4—101no—
279C2×D4D5⋊C811yescharacteristic
280C22×C4—21no—
281C22×C4—101no—
282C22×C4C22×AGL(1,𝔽5)11yescharacteristic
283Q16—54no—
284SD16—54no—
285SD16—54no—
286D8—54no—
287M16—21no—
288M16—52no—
289M16—52no—
290M16—52no—
291M16—102no—
292M16—102no—
293M16—102no—
294M16—102no—
295M16—101no—
296M16C22×AGL(1,𝔽5)12yes—
297C2×C8—21no—
298C2×C8—51no—
299C2×C8—51no—
300C2×C8—51no—
301C2×C8—51no—
302C2×C8—51no—
303C2×C8—52no—
304C2×C8—51no—
305C2×C8—102no—
306C2×C8—102no—
307C2×C8—102no—
308C2×C8—102no—
309C2×C8—101no—
310C4⋊C4—21no—
311C22⋊C4—21no—
312C42C22×AGL(1,𝔽5)12yes—
313C16—54no—
314C16—101no—
315C16—101no—
316C16—102no—
317C16—102no—
318C2×C5—21no—
319C2×C5—21no—
320C2×C5—21no—
321C2×C5—42no—
322C2×C5C2×C8×D411yescharacteristic
323D5—42no—
324C23—21no—
325C23—101no—
326C23#413111yescharacteristic
327Q8—21no—
328Q8—52no—
329Q8—101no—
330Q8C2×D5⋊C812yes—
331D4—22no—
332D4—22no—
333D4—21no—
334D4—52no—
335D4—102no—
336D4—102no—
337D4—101no—
338D4C2×D5⋊C812yes—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—101no—
344C2×C4—102no—
345C2×C4—102no—
346C2×C4C2×D5⋊C811yescharacteristic
347C2×C4C2×D5⋊C811yescharacteristic
348C2×C4#413111yescharacteristic
349C2×C4D4×AGL(1,𝔽5)11yescharacteristic
350C2×C4D4×AGL(1,𝔽5)11yescharacteristic
351C2×C4C23×AGL(1,𝔽5)11yescharacteristic
352C8—22no—
353C8—54no—
354C8—51no—
355C8—51no—
356C8—51no—
357C8—51no—
358C8—102no—
359C5#3344211yescharacteristic, complemented, Sylow
360C22—22no—
361C22—41no—
362C22—81no—
363C22—102no—
364C22—401no—
365C22#6569211yescharacteristic
366C22#6569211yescharacteristic
367C22C2×#413111yescharacteristic
368C4—22no—
369C4—42no—
370C4—102no—
371C4#6569211yescharacteristic
372C4#6569211yescharacteristic
373C4C22×D5⋊C811yescharacteristic
374C4C2×D4×AGL(1,𝔽5)11yescharacteristic
375C2—21no—
376C2—21no—
377C2—21no—
378C2—42no—
379C2—202no—
380C2C2×#6569211yescharacteristic
subgroupG/H
centerC4C2×D4×AGL(1,𝔽5)
commutatorC4×C5C23×C8
FrattiniC2×C4C23×AGL(1,𝔽5)
FittingC5×#421C4
socleC2×C5C2×C8×D4
prime radical#66156C4
cosocleC2×#168C24
hypercenter#421AGL(1,𝔽5)
hypocenterC5#33442
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #10617530C2semidirect
2. #10617404C2semidirect
4. C2×#64932C2semidirect
5. C2×#64898C2semidirect
6. #10604370C2semidirect
7. #10604366C2semidirect
8. #10604000C2semidirect
9. #10603999C2semidirect
14. #65791C22semidirect
21. #65725C22semidirect
29. #64932C22semidirect
30. #64932C22semidirect
31. #64898C22semidirect
32. #64898C22semidirect
33. #64861C22semidirect
34. #64859C22semidirect
35. C4×#674C22semidirect
359. C5#33442semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#106175302
#106174042
#106107202
C2×#649322
C2×#648982
#106043702
#106043662
#106040002
#106039992
#334425